FIRST TASK:

C. CHEMISTRY AND PHYSICS - ENTROPY One of the important concepts in chemistry and physics is the concept of entropy. Simply put, entropy gives us the idea that nature tends to go from order to disorder. This however has something to do with the arrangements of the particles in space. In this section. the connection of entropy with statistics, specifically the binomial distribution, will be explored. Suppose you have four gas particles in a chamber with two pans, left and right. Over time, the gas particles freely move about the chamber, and there are sixteen possible arrangements of the four gas particles, which can be grouped into: q? L3 .__r _.') l. Allontheleft ..-. i; , .~_, [1. Three on the left one on the right A? '3' n" "'_ 'o' \"' [ILTtm on each side '9' v. r "3" 4} '13:) W. Three on the right and two on the left a?\" J} 3" -.;-i c) :97 V. All on the right JiJ u');; 3- 4:) :3 -.:: .JM'} Each of these 16 possibilities is called a microstate, and each configuration is called a state. So, in the case of four particles, the state with the highest number of microstates is 111. This means that the most probable arrangement of the gas molecules is two on each side of the chamber. The probability of each state can be modelled using a binomial distribution. Suppose there are 4 particles, each of which have an equal chance of being on the leit chamber and on the right chamber. and each particle is independent of other particles. If X is the number of particles on the right pan of the chamber. then X will have a binomial distribution with n = 4 and p = 0.5. In the case that there are [00 particles in the chamber, and X is the number of molecules on the right pan of the chamber. 1. What is the most probable an'angement of gas molecules in the chamber? What is the probability of such arrangement? 2. What is the probability that 40 - 60 molecules are in the right chamber? 3. On MS Excel, list down all the numbers from 0 to 100 in one column A. On column B. compute the probability that X takes on the number on column A, 0.3. if 0 is on cell AI. then P[D) should be on cell BI. Use the BINOMDIST function to make it easier. Just enter the appropriate arguments, for X, rt. and p. Enter FALSE for the argument on whether you should get the cumulative distribution (last argument). Create a bar graph with the values on column B. Describe the shape of the bar graph. 4. Compute the mean and Standard deviation of X. 5. Using what you have learned from the first quarter, what is the probability that the normal distribution with mean and standard deviation equal to your answer to #4 has a value between 39.5 and 60.5? Compare your answer with your answer in #2